# 722edo

 ← 721edo 722edo 723edo →
Prime factorization 2 × 192
Step size 1.66205¢
Fifth 422\722 (701.385¢) (→211\361)
Semitones (A1:m2) 66:56 (109.7¢ : 93.07¢)
Dual sharp fifth 423\722 (703.047¢)
Dual flat fifth 422\722 (701.385¢) (→211\361)
Dual major 2nd 123\722 (204.432¢)
Consistency limit 5
Distinct consistency limit 5

722 equal divisions of the octave (abbreviated 722edo or 722ed2), also called 722-tone equal temperament (722tet) or 722 equal temperament (722et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 722 equal parts of about 1.66 ¢ each. Each step represents a frequency ratio of 21/722, or the 722nd root of 2.

722edo is a strong 2.7.19.23 subgroup tuning, with 179\722 being a semiconvergent to the log2(19/16). Despite having a strong approximation of 7, it is only consistent upwards to the 5-limit.

Using the 421\722 fifth, it supports a variant of fifth-stacked tuning that divides 38th harmonic into 9 parts, meaning that C - D# in this system is equal to 19/16, the otonal minor third. This creates a peculiar violation of Western theory which would require spelling this minor triad involving 19/16 as C-D#-G instead of C-Eb-G. This can be realized as 355 & 722 2.17.19.23 temperament from a regular temperament theory perspective - it should be noted that the fifth is not mapped to 3/2 but is slightly flatter.

Aside from this, 722bc val tempers out the hemifamity comma and is a tuning for the undecental temperament. Since 722 is divisible by 19, the 722dg val is a tuning for the kalium temperament in the 19-limit.

### Odd harmonics

Approximation of odd harmonics in 722edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.570 -0.718 +0.149 +0.522 +0.483 +0.470 +0.374 -0.246 -0.006 -0.421 -0.019
Relative (%) -34.3 -43.2 +9.0 +31.4 +29.0 +28.3 +22.5 -14.8 -0.4 -25.3 -1.2
Steps
(reduced)
1144
(422)
1676
(232)
2027
(583)
2289
(123)
2498
(332)
2672
(506)
2821
(655)
2951
(63)
3067
(179)
3171
(283)
3266
(378)

### Subsets and supersets

722edo has subset edos 1, 2, 19, 38, 361.